Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule

TL;DR

This paper addresses the distortion of multi-winner elections in a line metric under additive social costs. It introduces the Polar Comparison Rule, leverages a focal-point voter-movement framework, and proves sharp distortion bounds: at most $2.33$ for all $k>2$, with tight constants $2.41$ for $k=2$ and $2.33$ for $k=3$; it further provides parity-based and asymptotic lower bounds and extends results to egalitarian additive cost. The approach combines candidate-ordering on a line, obstacle-based movement arguments, and rule-composition to obtain tight results across all $k$, including a general bound near $7/3$ for many $k$, and a $k$-Extremes bound in the egalitarian setting. These findings advance understanding of how ordinal information can yield nearly optimal committee selection in metric environments and suggest directions for broader metric spaces and tighter bounds.

Abstract

We study the problem of minimizing metric distortion in multi-winner elections, where a committee of size $k$ is selected from a set of candidates based on voters' ordinal preferences. We assume that voters and candidates are embedded on a line metric, and social cost is determined by the underlying metric distances. The distortion of a voting rule is the worst-case ratio between the social cost of the elected committee and an optimal committee. Previous work has focused on the $q$-cost model, in which a voter's cost is given by the distance to their $q$th closest committee member. Here, we study the additive cost, where a voter's cost is the sum of distances to all committee members. We introduce the Polar Comparison Rule and analyze its distortion under utilitarian additive cost. We show that it achieves a distortion of at most $2.33$ for all committee sizes $k>2$, improving upon the previously best-known upper bound of $3$. Moreover, for $k=2$ and $k=3$, we establish tight distortion bounds of $2.41$ and $2.33$, respectively. We also derive lower bounds that depend on the parity of $k$ and analyze the behavior of distortion for small and large committee sizes. Finally, we extend our results to the egalitarian additive cost.

Figures (6)

• Figure 1: An instance with $n$ voters and $9$ candidates. The optimal committee of size $k = 3$ depends on how voter cost is defined in terms of the committee members.
• Figure 2: If $a$ and $b$ are the top two candidates in the Majority order and $\{o_1, o_2\}$ is the optimal committee of size 2, by assuming that $a = o_1$ and moving the location of voters, one can prove that $\mathrm{dist}(\{a, b\}) \le 2$.
• Figure 7: Instance $\mathcal{E}$ with two metrics $d_1$ (a) and $d_2$ (b), used for proving the lower bound in $2$-winner election. $a, b, a',$ and $b'$ are the candidates distributed on locations $-1$ and $+1$.
• Figure 10: Instance $\mathcal{E}$ with two metrics, $d_1$ (a) and $d_2$ (b), used to prove the lower bound on the distortion of $k$-winner voting rules for $k \le m/2$, as discussed in Theorem \ref{['thm:lb-general']}. $S_a$ and $S_b$ are two subsets of candidates located at positions $-1$ and $+1$. As shown in the figures, voters are divided into two groups of sizes $\lceil n/(1+\alpha)\rceil$ and $\lfloor n\alpha/(1+\alpha)\rfloor$.
• Figure 11: Instance $\mathcal{E}$ with two metrics, $d_1$ (a) and $d_2$ (b), used to prove the lower bound on the distortion of $k$-winner voting rules for $k > m/2$, as discussed in Theorem \ref{['thm:lb->m/2']}. $S_a$ and $S_b$ are two subsets of candidates located at positions $-1$ and $+1$. As shown in the figures, voters are divided into two groups of size $n/2$.

Theorems & Definitions (35)

• Lemma 1
• proof
• proof : Proof of Lemma \ref{['lem:candidates-order']}
• Lemma 2
• proof
• Corollary 1: of Lemma \ref{['lem:relative-sc']}
• Lemma 3
• proof
• Lemma 4: ANSHELEVICH201827
• Corollary 2: of Lemma \ref{['lem:ratio-alters']}